International Mathematics Competition
for University Students
2026

IMC2026: Day 2, Problem 6

Problem 6.

(a) Is there a differentiable function \(\displaystyle f\colon \mathbb{R} \to \mathbb{R}\) such that f'(f(x)) = x for every \(\displaystyle x \in \mathbb{R}?\)

(b) Is there a differentiable function \(\displaystyle f\colon \mathbb{R} \to \mathbb{R}\) such that f'(f(x)) = |x| for every \(\displaystyle x \in \mathbb{R}?\)

Nikolaos Kolliopoulos, University of Cyprus

  Solution